紧Kahler流形上的复Hessian方程

Complex Hessian equations with prescribed singularity on compact Kähler manifolds

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$ and fix $1\leq m\leq n$. We prove that the total mass of the complex Hessian measure of $ω$-$m$-subharmonic functions is non-decreasing with respect to the singularity type. We then solve complex Hessian equations with prescribed singularity, and prove a Hodge index type inequality for positive currents.

紧致凯勒流形上二黑塞方程的一个数值准则

A Numerical Criterion for the 2-Hessian Equation on Compact Kähler Manifolds

研究紧致凯勒流形上二黑塞方程，通过相关准则产生高杜雄类，复三维时与可解性等价，据此证明了村上和塞克利希迪的相关猜想，并建立了边界版本。

紧致Kähler流形上Hessian型方程的Nakai-Moishezon型准则

Nakai-Moishezon type criteria for Hessian type equations on compact Kähler manifolds

本文为紧致Kähler流形上的Hessian型方程建立了Nakai-Moishezon型数值准则，并应用于证明广义Monge-Ampère、复Hessian、Hessian商及临界LYZ方程解的存在唯一性，同时刻画了$k$-正类锥的结构。

凯勒流形上热方程正解的黑塞矩阵的上界

Upper Bounds for Hessian Matrices of Positive Solutions to Heat Equations on Kähler Manifolds

研究凯勒流形上热方程正解的黑塞矩阵上界，通过证明全局和局部上界及改进韩和张的结果，将双边曲率界下的上界扩展到截面曲率有下界的黎曼流形。

Hessian equations on closed Hermitian manifolds

In this paper, using the technical tools in \cite{TW5}, we solve the complex Hessian equation on closed Hermitian manifolds, which generalizes the the Kähler case results in \cite{HMW} and \cite{DK}.

Nakai-Moishezon criterions for complex Hessian equations

The $J$-equation proposed by Donaldson is a complex Hessian quotient equation on Kähler manifolds. The solvability of the $J$-equation is proved by Song-Weinkove to be equivalent to the existence of a subsolution. It is also conjectured by Lejmi-Szekelyhidi to be equivalent to a stability condition in terms of holomorphic intersection numbers as an analogue of the Nakai-Moishezon criterion in algebraic geometry. The conjecture is recently proved by Chen under a stronger uniform stability condition. In this paper, we establish a Nakai-Moishezon type criterion for pairs of Kähler classes on anal…

关于横截 Kähler 流形上复 Hessian 方程 $L^\infty$ 估计的 PDE 方法注记

A note on the PDE approach to the $L^\infty$ estimates for complex Hessian equations on transverse Kähler manifolds

本文改编 Guo-Phong-Tong 等人的 PDE 方法，证明横截 Kähler 流形上复 Monge-Ampère 方程的 $L^\infty$ 估计，并由此给出 Q-Gorenstein T-簇上 Calabi-Yau 锥度量正则性的纯 PDE 证明。

复Hessian方程梯度估计的另一种证明

Alternative proof of the gradient estimate for complex Hessian equations

本文针对紧致及带边界的Kähler流形上的复Hessian方程，提出一种无需爆破论证的梯度估计新证明，并明确其与右端项的关系。

紧致凯勒流形上具有规定奇异性的Hessian型方程的解

Solution to Hessian type equations with prescribed singularity on compact Kahler manifold

本文将Darvas-DiNezza-Lu的相对多重位势理论推广到Hessian框架，刻画了Hessian算子的有限能量范围，并证明在自然条件下退化复Hessian方程的解与给定模型位势具有相同的奇异性类型。

Kähler流形上的特殊Ricci-Hessian方程

Special Ricci-Hessian equations on Kähler manifolds

本文研究Kähler流形上的特殊Ricci-Hessian方程，揭示其在复维数二时的三种类型，并通过Cartan-Kähler定理证明其非标准实现方式。

正常 Hessian 截面曲率的 Hessian 流形

Hessian manifolds of positive constant Hessian sectional curvature

本文分类了正常 Hessian 截面曲率的 Hessian 流形标准模型，通过引入 Kähler 流形在交换李群 Hamilton 作用下的商结构实现该分类，填补了正曲率情形的空白。

Projective Hessian and Sasakian manifolds

The Hessian geometry is the real analogue of the Kähler one. Sasakian geometry is an odd-dimensional counterpart of the Kähler geometry. In the paper, we study the connection between projective Hessian and Sasakian manifolds analogous to the one between Hessian and Kähler manifolds. In particular, we construct a Sasakian structure on $TM\times \mathbb{R}$ from a projective Hessian structure on $M$. Especially, we are interested in the case of invariant structure on Lie groups. We define semi-Sasakian Lie groups as a generalization of Sasakian Lie groups. Then we construct a semi-Sasakian struc…

C^0 estimates for Hessian quotient equations on HKT manifolds

We show the $C^0$ estimate for solutions to Hessian quotient equations on hyperKähler with torsion manifolds without any additional assumption on its hypercomplex structure by a clever use of the cone condition directly.

关于多项式复Hessian方程I：黎曼流形上的非线性非谱椭圆Hessian方程

On Polynomial Complex Hessian Equations I: Non-Linear Non-Spectral Elliptic Hessian Equations on Riemannian Manifolds

本文研究紧致黎曼流形上的一般完全非线性椭圆Hessian方程，通过弱化标准对称性假设，在推导先验估计后利用连续性方法求解。

辅助Hessian方程

Auxiliary Hessian Equations

该研究提出k-Hessian方程作为Kähler类退化为数值维数k的nef类的辅助方程，给出Hessian方程的一致均值不等式与先验估计，为相关退化问题提供概念验证结果。

The Dirichlet problem for mixed Hessian equations on Hermitian manifolds

In this paper we study the Dirichlet problem for a class of Hessian type equation with its structure as a combination of elementary symmetric functions on Hermitian manifolds. Under some conditions with the initial data on manifolds and admissible subsolutions, we derive a priori estimates for this complex mixed Hessian equation and solvability of the corresponding Dirichlet problem.

关于Hermitian流形上涉及Hessian算子之和的方程的二阶估计

Second order estimates for equations with sums of Hessian operators on Hermitian manifolds

本文基于动态 plurisubharmonic 条件，为满足条件的可允许解建立二阶估计，利用复Hessian算子之和的凹性不等式推导。

On the Hessian-cscK equations

In this paper, we propose a coupled system of complex Hessian equations which generalizes the equation for constant scalar curvature Kähler (cscK) metrics. We show this system can be realized variationally as the Euler-Lagrange equation of a Hessian version of the Mabuchi K-energy in an infinite dimensional space of $k$-Hessian potentials, which can be seen as an infinite dimensional Riemannian manifold with negative sectional curvature. Finally, we prove an a priori $C^0$-estimate for this system which depends on the Entropy, which generalizes a fundamental result of Chen and Cheng for cscK m…

A class of Hessian quotient equations in the warped product manifold

In this paper, we consider a class of Hessian quotient equations in the warped product manifold $\overline{M}=I\times_λM$. Under some sufficient conditions, we obtain an existence result for the star-shaped compact hypersurface $Σ$ in $\overline{M}$ using standard degree theory based on a priori estimates for solutions to the Hessian quotient equations.

Cusp Kähler-Ricci flow on compact Kähler manifold

In this paper, by limiting twisted conical Kähler-Ricci flows, we prove the long-time existence and uniqueness of cusp Kähler-Ricci flow on compact Kähler manifold $M$ which carries a smooth hypersurface $D$ such that the twisted canonical bundle $K_M+D$ is ample. Furthermore, we prove that this flow converge to a unique cusp Kähler-Einstein metric.

Convergence of the Hesse-Koszul flow on compact Hessian manifolds

We study the long time behavior of the Hesse-Koszul flow on compact Hessian manifolds. When the first affine Chern class is negative, we prove that the flow converges to the unique Hesse-Einstein metric. We also derive a convergence result for a twisted Hesse-Koszul flow on any compact Hessian manifold. These results give alternative proofs for the existence of the unique Hesse-Einstein metric by Cheng-Yau and Caffarelli-Viaclovsky as well as the real Calabi theorem by Cheng-Yau, Delanoë and Caffarelli-Viaclovsky.

三维Hessian流形的拓扑结构

The topology of 3-dimensional Hessian manifolds

本文研究了三维Hessian流形的拓扑结构，证明其为Hantzsche-Wendt流形或Kähler映射扭结构，并给出了完整的拓扑分类。

具有正第一Chern class的环面流形上的里奇流

Kähler-Ricci flow on a toric manifold with positive first Chern class

本文证明了在具有正第一陈类的$n$维紧致环面流形上，具有任意初始$(S^1)^n$不变Kähler度量的Kähler-里奇流收敛于Kähler-Ricci孤子.特别地，利用Kähler-里奇流给出了第一Chern类为正的紧致环面流形上Kähler-Ricci孤子存在性的另一个证明.

The Dirichlet Problem for the $k$-Hessian Equation on a complex manifold

We solve the Dirichlet problem for $k$-Hessian equations on compact complex manifolds with boundary, given the existence of a subsolution. Our method is based on a second order a priori estimate of the solution on the boundary with a particular gradient scale. The scale allows us to apply a blow-up argument to obtain control on all necessary norms of the solution.

Monge-Ampère equations on compact Hessian manifolds

We consider degenerate Monge-Ampère equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampère operators. We then use the Perron method to solve Monge-Ampère equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoë, Caffarelli-Viaclovsky and Hultgren-Önnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow…

Improved Beckner-Sobolev inequalities on Kähler manifolds

We prove new Beckner-Sobolev type inequalities on compact Kähler manifolds with positive Ricci curvature. As an application, we obtain a diameter upper bound that improves the Bonnet-Myers bound.

混合Hessian不等式与$\mathcal{E}(X,ω,m)$类的唯一性

Mixed Hessian inequalities and uniqueness in the class $\mathcal{E}(X,ω,m)$

利用整体论证证明了混合Hessian测度的一个一般不等式。对于复蒙日-安培方程的情形，我们的方法也得到了简化.利用这一点，并使用Kokiodziej的质量浓度技术，我们还证明了在概率测度消失的情况下，在$m$ -极集的紧凑Kähler流形上的复杂的Hessian方程的解决方案的唯一性。

复Hessian方程的黏性解

Viscosity solutions to complex Hessian equations

研究了复海森方程的黏性解。在局部情形下，我们认为$Ω$是$\mathbb{C}^n,$中的有界区域$β$是$\mathcal{C}^n$和$1\leq m\leq n.$中的标准Kähler形式.在$F, g$上的适当条件下，我们证明了在$\pO$上的方程$(dd^c φ)^m\wedgeβ^{n-m}=F(x,φ)β^n,\ \f=g$存在唯一的粘性解.此外，如果数据是霍德尔连续的，那么解也是如此。在整体情形下，令$(X,ω)$是紧致埃尔米特齐次流形，其中$ω$是不变埃尔米特度量（不一定是Kähler）。证明了方程$(ω+dd^cφ)^m\wedgeω^{n-m}=F(x,φ)ω^n$在某些自然条件下在$F.$上存在唯一的粘性解

The Dirichlet problem for a complex Hessian equation on compact Hermitian manifolds with boundary

We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal Kähler metrics, provided a subsolution.

Selfsimilar Hessian manifolds

A selfsimiar manifold is a Riemannian manifold $\left(M,g\right)$ endowed with a homothetic vector field $ξ$. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian manifold or a Eucledean space. A radiant Hessian manifold is selfsimilar Hessian manifold $\left(M,\nabla,g,ξ\right)$ such that $\nablaξ=λ\text{Id}$. We prove that any selfsimilar Hessian manifold with a potential homothetic vector field is locally isomorphic to a product radiant H…

扭量空间上的自然Hermitian型Hessian

Hessian of the natural Hermitian form on twistor spaces

我们依次计算了超Kähler流形的Calabi族、四维反自对偶黎曼流形的扭量空间和四元数Kähler流形的扭量空间上的自然Hermitian形式的hessian。我们证明了超凯勒流形的Calabi族上扭量线的循环空间的强凸性。我们还证明了非正数量曲率的四维反自对偶Einstein流形的扭量空间的1-圈空间和非正数量曲率的四元数Kähler流形的扭量空间的1-圈空间的凸性。我们检查，没有非Kähler强KT流形出现这样的扭量空间。

Second order estimates for a class of complex Hessian equations on Hermitian manifolds

In this paper, we derive an \emph{a priori} second order estimate for solutions which are in $Γ_{k+1}$ cone to a class of complex Hessian equations with both sides of the equation depending on the gradient on compact Hermitian manifolds.

关于Kahler流形的几个比较定理

Some comparison theorems for Kahler manifolds

在这项工作中，我们将验证Kahler流形上的一些比较结果。它们是Kahler流形的闭复子流形的距离函数的复Hessian比较，特征值比较和体积比较，其中Kahler流形的闭复子流形的全纯双截曲率下有界，特征值比较和体积比较是关于数量曲率的.本文的工作是由李和王的比较结果所激发的。

几乎Hermite流形的Hessian比较和eigenvalue估计

Hessian comparison and eigenvalue estimate of almost Hermitian manifolds

利用几乎Hermitian流形上的Bochner技巧，推广了Tossati关于几乎Hermitian流形的Laplacian比较，得到了几乎Hermitian流形的复Hessian比较，并改进了Gray关于几乎Hermitian流形的直径估计.此外，我们还得到了拟Kaehler流形上的一个尖锐的特征值估计和近似Kaehler流形上的一个尖锐的Hessian比较。

退化复Hessian方程的解

Solutions to degenerate complex Hessian equations

设$(X,ω)$是$n$维紧Kähler流形.研究了$(ω+dd^cφ)^m\wedge ω^{n-m}=F(x,φ)ω^n.$形式的退化复Hessian方程在$F$上的某些自然条件下，该方程存在唯一的连续解.当$(X,ω)$有理齐次时，我们进一步证明了解是Hölder连续的.

A generalised Ricci-Hessian equation on Riemannian manifolds

In this paper, we prove new rigidity results related to some generalised Ricci-Hessian equation on Riemannian manifolds.

复Hessian方程的光滑解

Smooth solutions to the complex Hessian equation

设$(X,ω)$是一个维数为$n$的紧致Kähler流形，且固定$1\leq m\leq n.$我们证明了复Hessian方程$(ω+dd^cφ)^m\wedge ω^{n-m}=fω^n$，具有$0<f\in \mathcal{C}^{\infty}(X)$，有光滑的可采纳解$ φ\in \mathcal{C}^{\infty}(X)$ .以前已知当$(X,ω)$具有非负的全纯对分曲率时，这是成立的。

On uniform estimate of complex elliptic equations on closed Hermitian manifolds

In this paper, we study Hessian equations and complex quotient equations on closed Hermitian manifolds. We directly derive the uniform estimate for the admissible solution. As an application, we solve general Hessian equations on closed Kähler manifolds.

A splitting theorem for Riemannian manifolds of generalised Ricci-Hessian type

In this paper, we study and partially classify those Riemannian man-ifolds carrying a non-identically vanishing function f whose Hessian is minus f times the Ricci-tensor of the manifold.

Kahler流形中的H-极小Lagrange纤维化与Kahler-Einstein流形中的极小Lagrange消失环面

H-minimal Lagrangian fibrations in Kahler manifolds and minimal Lagrangian vanishing tori in Kahler-Einstein manifolds

一般Kähler流形中的H-极小拉格朗日子流形推广了Calabi-Yau流形中的特殊拉格朗日子流形.本文利用Kähler流形中H-极小拉格朗日子流形的形变理论，构造了具有负第一Chern类的Kähler-Einstein流形中的极小Lagrange环面.

Hess流形上几何流的孤子

On solitons of a geometric flow on Hessian manifolds

本文针对Hess流形上的几何流，引入Koszul孤子并研究其方程、性质与实例，通过关联仿射流形与复流形的方法拓展了几何问题的研究，该孤子在统计学与信息几何中具应用潜力。

Vanishing theorems of $L^2$-cohomology groups on Hessian manifolds

We show vanishing theorems of $L^2$-cohomology groups of Kodaira-Nakano type on complete Hessian manifolds. We obtain further vanishing theorems of $L^2$-cohomology groups $L^2H^{p,q}(Ω)$ on a regular convex cone $Ω$ with the Cheng-Yau metric for $p>q$.

Hessian Operators on Constraint Manifolds

On a constraint manifold we give an explicit formula for the Hessian matrix of a cost function that involves the Hessian matrix of a prolonged function and the Hessian matrices of the constraint functions. We give an explicit formula for the case of the orthogonal group ${\bf O}(n)$ by using only Euclidean coordinates on $\mathbb{R}^{n^2}$. An optimization problem on ${\bf SO}(3)$ is completely carried out. Its applications to nonlinear stability problems are also analyzed.

关于非紧 Hessian 流形的拓扑刚性定理

A topological rigidity theorem on noncompact Hessian manifolds

本文通过在非紧仿射黎曼流形上构造几何流的短时间解，建立了具有非负 Hessian 截曲率的 Hessian 度量，并证明了切丛具有最大体积增长的完备非紧 Hessian 流形微分同胚于 $\mathbb{R}^n$。

射影流形上复Hessian型方程的数值准则

A numerical criterion for complex Hessian type equations on projective manifolds

本研究针对射影流形上的复Hessian型方程，证明了对应于强严格右诺特多项式及任意次数严格右诺特多项式的Nakai-Moishezon型准则，覆盖复Hessian方程等相关方程类。

关于紧致Hessian流形的拓扑学

On Compact Hessian Manifolds

本文研究了紧致Hessian流形的全局拓扑约束和结构性质，通过新建立的纤维化和分解定理，验证了Chern关于此类仿射流形欧拉特征消失的猜想，并利用Hitchin系统理论和Cheng-Yau解实Monge-Ampère方程，对闭合定向Hessian3流形进行几何分类。

Hessian几何与具有曲率的Frobenius流形

Hessian geometry and Frobenius manifolds with curvature

本文研究了具有曲率的Frobenius流形的几何结构，证明了其与Hessian结构的一致性，并展示了其在超积分系统中的应用。

Warped Product Conformal Kahler Manifolds and Kenmotsu Structures

In this paper, we investigate the condition on warped product manifold of a Kenmotsu manifold and the real line to be a conformal Kahler manifold. This result demonstrates the close relation of Kenmotsu and Kahler manifolds.

Relations between the Kahler cone and the balanced cone of a Kahler manifold

In this paper, we consider a natural map from the Kahler cone to the balanced cone of a Kahler manifold. We study its injectivity and surjecticity. We also give an analytic characterization theorem on a nef class being Kahler.

Hessian流形的对偶性与最优传输

Duality of Hessian manifolds and optimal transport

本文综述了Hessian流形对偶理论在最优传输中的应用，用于求解Monge-Ampère方程，并探讨其在镜像对称SYZ猜想中的最新进展。

Hessian of Busemann functions and rank of Hadamard manifolds

In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold $(M,g)$ corresponding to eigenvalue zero is investigated with respect to rank of geodesics. On a harmonic Hadamard manifold which is of purely exponential volume growth, or of hypergeometric type it is shown that every Busemann function admits positive definite Hessian. A criterion for $(M,g)$ fulfilling …

Rigidity theorems of complete Kähler-Einstein manifolds and complex space forms

We derive some elliptic differential inequalities from the Weitzenböck formulas for the traceless Ricci tensor of a Kähler manifold with constant scalar curvature and the Bochner tensor of a Kähler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some $L^p$ and $L^\infty $ pinching results are established to characterize Kähler-Einstein manifolds among Kähler manifolds with constant scalar curvature, and others are given to characterize complex space forms among Kähler-Einstein manifolds. Finally, these pinching results may be combined to characterize complex spa…

流形上的高阶Hessian结构

Higher order Hessian structures on manifolds

本文定义了流形上的$n$阶Hessian结构并对其进行了研究.特别地，当$n = 3$时，我们对流形的二阶切丛上的{\it三阶Hessian结构}和{\it某类联络}进行了详细的研究并建立了一一对应.进一步，我们证明了流形的切丛上的联络诱导二阶切丛上的联络。定义了特殊二阶联络的二阶测地线，给出了对称三阶Hessian结构的一个几何刻画。

具有非负双正交曲率的近凯勒四维流形

Almost-Kähler four manifolds with nonnegative biorthogonal curvature

本文对具有非负双正交曲率的紧致近凯勒四维流形进行了完整分类。